Neighborly translational tessellations of the n-torus

Abstract

The concept of an n-NTT (neighborly translational tessellation of the n-torus) is introduced as a tessellation where every pair of tiles are translates of each other, and share precisely one of their facets. An n-NTT with cubic tiles is studied for each \(n \in \mathbb {N}\), and particular attention is given to a 4-NTT whose tiles are isometric 24-cells. We also use this concept to describe a tessellation of \(\mathbb {E}^4\) with isometric tiles with fractal boundary, as well as a NTT of an infinite-dimensional space.